最適化アルゴリズムのテスト関数 (1)

最適化アルゴリズムの性能の良し悪しを決めるために，テスト関数 (ベンチマーク関数) が用いられる．テスト関数は，探索の範囲が設定され，最小値 (最適値) が既知であるような関数である．

この資料では，最適化アルゴリズムの評価に用いるテスト関数をプロットする．テスト関数としては，英語版の Wikipedia[1] に載っているものを用いる．

[1]: Test functions for optimization (en:Wikipedia), Accessed on 2021/09/09.

########## Packages ##########
import numpy as np
from lib.lib import Test_function
from lib.plot import Plot_func
##############################

##### instance of the class #####
test = Test_function()
pf = Plot_func()
#################################

Rastrigin function

Definition

$$f(\boldsymbol{x}) = An + \sum_{i = 1}^n (x_i^2 - A \cos (2 \pi x_1)), \ \ \text{where} \ \ A = 10.$$

Search domain

$$ -5.12 \leq x_i \leq 5.12, \ \ i = 1, \ldots, n. $$

Global minimum

$$ f(0, \ldots, 0) = 0 $$

##### Plot #####
x = np.linspace(-5.12, 5.12, 100)
y = np.linspace(-5.12, 5.12, 100)
pf.plotly_surface_contour(func=test.rastrigin, 
                    x_range=x, y_range=y, func_name="rastrigin")
################

Ackley function

Definition

$$ f(x, y) = -20 \exp \left(-0.2 \sqrt{0.5(x^2 + y^2)} \right) - \exp \left(0.5 (\cos 2 \pi x + \cos 2 \pi y)\right) + e + 20.$$

Search domain

$$ -5 \leq x, y \leq 5. $$

Global minimum

$$ f(0, 0) = 0 $$

##### Plot #####
x = np.linspace(-5, 5, 100)
y = np.linspace(-5, 5, 100)
pf.plotly_surface_contour(func=test.ackley, 
                    x_range=x, y_range=y, func_name="ackley")
################

Sphere function

Definition

$$ f(\boldsymbol{x}) = \sum_{i = 1}^n x_i^2. $$

Search domain

$$ - \infty \leq x_i \leq \infty, \ \ i = 1, \ldots, n. $$

Global minimum

$$ f(0, \ldots, 0) = 0. $$

#### Plot #####
x = np.linspace(-5, 5, 100)
y = np.linspace(-5, 5, 100)
pf.plotly_surface_contour(func=test.sphere, 
                    x_range=x, y_range=y, func_name="sphere")
###############

Rosenbrock function

Definition

$$ f(\boldsymbol{x}) = \sum_{i = 1}^{n - 1} \left( 100(x_{i + 1} - x_i^2)^2 + (1 - x_i)^2 \right). $$

Search domain

$$ -\infty \leq x_i \leq \infty, \ \ i = 1, \ldots, n. $$

Global minimum

$$ f(1, \ldots, 1) = 0.$$

#### Plot #####
x = np.linspace(-0.5, 1.25, 100)
y = np.linspace(-0.5, 1.25, 100)
pf.plotly_surface_contour(func=test.rosenbrock, 
                    x_range=x, y_range=y, func_name="rosenbrock")
###############

Beale function

Definition

$$f(x, y) = (1.5 - x + xy)^2 + (2.25 - x + xy^2)^2 + (2.625 - x + xy^3)^2.$$

Search domain

$$-4.5 \leq x, y \leq 4.5.$$

Global minimum

$$f(3, 0.5) = 0.$$

#### Plot #####
x = np.linspace(-4.5, 4.5, 100)
y = np.linspace(-4.5, 4.5, 100)
pf.plotly_surface_contour(func=test.beale, x_range=x, y_range=y, func_name="beale")
###############

Goldstein-Price function

Definition

$$ f(x, y) = [1 + (x + y + 1)^2(19 - 14x + 3x^2 - 14y + 6xy + 3y^2)]$$$$\times[30 + (2x - 3y)^2(18 - 32x + 12x^2 + 48y - 36xy + 27y^2)]. $$

Search domain

$$-2 \leq x, y \leq 2.$$

Global minimum

$$ f(0, -1) = 3. $$

#### Plot #####
x = np.linspace(-2, 2, 100)
y = np.linspace(-2, 2, 100)
pf.plotly_surface_contour(func=test.goldstein_price, 
                    x_range=x, y_range=y, func_name="goldstein_price")
###############

Booth function

Definition

$$f(x, y) = (x + 2y - 7)^2 + (2x + y - 5)^2.$$

Search domain

$$-10 \leq x, y \leq 10.$$

Global minimum

$$ f(1, 3) = 0 $$

#### Plot #####
x = np.linspace(-10, 10, 100)
y = np.linspace(-10, 10, 100)
pf.plotly_surface_contour(func=test.booth, 
                    x_range=x, y_range=y, func_name="booth")
###############

Bukin function N.6

Definition

$$ f(x, y) = 100 \sqrt{|y - 0.01x^2|} + 0.01 |x + 10|. $$

Search domain

$$-15 \leq x \leq 5, \ -3 \leq y \leq 3.$$

Global minimum

$$ f(-10, 1) = 0. $$

#### Plot #####
x = np.linspace(-15, 5, 100)
y = np.linspace(-3, 3, 100)
pf.plotly_surface_contour(func=test.bukin, 
                    x_range=x, y_range=y, func_name="bukin")
###############

Matyas function

Definition

$$ f(x, y) = 0.26(x^2 + y^2) - 0.48xy. $$

Search domain

$$ -10 \leq x, y \leq 10. $$

Global minimum

$$ f(0, 0) = 0. $$

#### Plot #####
x = np.linspace(-10, 10, 100)
y = np.linspace(-10, 10, 100)
pf.plotly_surface_contour(func=test.matyas, 
                    x_range=x, y_range=y, func_name="matyas")
###############

Levi function N.13

Definition

$$ f(x, y) = \sin^2 3 \pi x + (x - 1)^2 (1 + \sin^2 3 \pi y) $$$$ + (y - 1)^2 (1 + \sin^2 2 \pi y). $$

Search domain

$$ -10 \leq x, y \leq 10.$$

Global minimum

$$f(1, 1) = 0.$$

#### Plot #####
x = np.linspace(-10, 10, 100)
y = np.linspace(-10, 10, 100)
pf.plotly_surface_contour(func=test.levi, 
                    x_range=x, y_range=y, func_name="levi")
###############